The multicolor book graph size Ramsey number conjecture

Let $B_n^{(k)}$ denote the book graph consisting of $k$ triangles sharing a common edge, with the shared edge contained in $k$ pages, each having $n$ vertices in the relevant book parameter as used in the paper. For a graph $H$ and integer $q\geq 2$, let $\hat r(H;q)$ be the minimum number of edges in a graph such that every $q$-edge-coloring contains a monochromatic copy of $H$, and let $r(K_k;q-1)$ be the $(q-1)$-color Ramsey number of the complete graph $K_k$. Multicolor book graph size Ramsey conjecture. Fix $q\geq 3$. For every $k\geq 2$ and all sufficiently large $n$,

$$\hat r(B_n^{(k)};q)=\Theta\left(q^k\cdot n^2\cdot r(K_k;q-1)\right).$$

This extends the paper's two-color book-graph results to fixed multicolor settings; the conjecture is presented as an open direction, and the required order of magnitude is not established in the source.